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\def\calR{{\mathcal R}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\calV{{\mathcal V}}
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\def\frakg{{\mathfrak g}}

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\def\eps{\epsilon}
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\begin{document}
\parindent 0in

\parbox[b]{1.5in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2122}{2021-22}: \href{http://drorbn.net/21-1350}{MAT 1350 Knot Theory}:
}~{\LARGE\bf CU and QU}
\hfill\parbox[b]{2in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill \today
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/21-1350}
}~\parbox[b]{0.3in}{\tiny\href{http://www.math.toronto.edu/~drorbn/bv.html}{Better\newline vaxxed!}}

\vskip -3mm
\rule{\linewidth}{1pt}
\vspace{-8mm}

\begin{multicols}{2}

\begin{definition} Let $\sleps$
be the Lie algebra with generators $\{\by,\bb,\ba,\bx\}$ and with commutation relations
\[
  [\ba,\bx]=\bx,\quad
  [\bb,\by]=-\epsilon\by,\quad
  [\ba,\bb]=0,\quad
  [\ba,\by]=-\by,
\]
\[
  [\bb,\bx]=\epsilon\bx,\quad
  [\bx,\by]=\bb+\epsilon\ba.
\]
\end{definition}

\begin{definition} Let $CU\coloneqq \calU(\sleps)$ be the universal
enveloping algebra of $\sleps$. Namely, $CU$ is the associative algebra
$A\langle \by,\bb,\ba,\bx\rangle$ generated by the same $\{\by,\bb,\ba,\bx\}$, subject to
the same relations as above. We denote the multiplication map of $CU$ by 
$cm^{ij}_k\colon CU_i\otimes CU_j\to CU_k$. $CU$ is a Hopf
algebra in the standard way; namely, with its given associative algebra
structure and with unit $c\eta\colon\bbQ\to CU$, counit $c\veps\colon
CU\to\bbQ$, antipode $cS\colon CU\to CU$, and coproduct
$c\Delta\colon CU\to CU\otimes CU$ given as follows:
\[ \begin{aligned}
  c\eta_i(\lambda) &= \lambda\cdot 1_i, \\
  c\veps^i(1_i,y_i,b_i,a_i,x_i) &= (1,0,0,0,0), \\
  cS_i(y_i,b_i,a_i,x_i) &= (-y_i,-b_i,-a_i,-x_i), \\
  c\Delta^i_{jk}(y_i,b_i,a_i,x_i) &=
    (y_j+y_k, b_j+b_k, a_j+a_k, x_j+x_k).
\end{aligned} \]
\end{definition}

\begin{definition}
Let $QU$, a ``quantization'' of $CU$, be the associative algebra $A\langle
\by,\bb,\ba,\bx\rangle\llbracket\hbar\rrbracket$ over the ring $\bbQ\llbracket\hbar\rrbracket$ modulo to the relations
\[
  [\ba,\bx]=\bx,\quad
  [\bb,\by]=-\epsilon\by,\quad
  [\ba,\bb]=0,\quad
  [\ba,\by]=-\by,
\]
\[
  [\bb,\bx]=\epsilon\bx,\quad
  \bx\by-q\by\bx=\frac{1-\bA\bB}{\hbar},
\]
where $q\coloneqq\bbe^{\hbar\epsilon}$, $\bA\coloneqq\bbe^{-\hbar\eps\ba}$, and $\bB\coloneqq\bbe^{-\hbar\bb}$.
We denote the multiplication map of $QU$ with $qm^{ij}_k\colon QU_i\otimes QU_j\to QU_k$. We also set
\[ \begin{aligned}
  q\eta_i(\lambda) &= \lambda\cdot 1_i, \\
  q\veps^i(1_i,\by_i,\bb_i,\ba_i,\bx_i) &= (1,0,0,0,0), \\
  qS_i(\by_i,\bb_i,\ba_i,\bx_i) &= (-\bB_i^{-1}\by_i,-\bb_i,-\ba_i,-\bA_i^{-1}\bx_i), \\
  q\Delta^i_{jk}(\by_i,\bb_i,\ba_i,\bx_i) &= (\by_j \!+\! \bB_j\by_k,\bb_j \!+\! \bb_k, \ba_j \!+\! \ba_k, \bx_j \!+\! \bA_j\bx_k).
\end{aligned} \]
\end{definition}

\begin{definition} \label{def:R}
Let $R_{ij}$ be the element of $QU_i\otimes QU_j$ given by the following formula:
\[ R_{ij}
  = \sum_{m,n\geq 0}\frac{\by_i^n\bb_i^m(\hbar \ba_j)^m(\hbar \bx_j)^n}{m![n]_q!}
  \sim \bbe^{\hbar \bb_i\ba_j}\bbe_q^{\hbar\by_i\bx_j},
\]
where\hfill$\displaystyle [n]_q!\coloneqq[1]_q[2]_q\cdots[n]_q,$\hfill\null
\[ [k]_q \coloneqq 1+q+q^2+\ldots+q^{k-1} = \frac{q^k-1}{q-1}, \]
and\hfill$\displaystyle \bbe_q^z \coloneqq \sum_{n=0}^\infty\frac{z^n}{[n]_q!}.$\hfill\null
\end{definition}

\begin{convention} For both $CU$ and $QU$, we always order the generators in the order $\by\bb\ba\bx$.
\end{convention}

\noindent{\bf Questions.}
\begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-4pt,topsep=0pt]
\item Why call $\sleps$ that way?
\item How do we compute in $CU$?
\item How did $QU$ come about? How is it a ``quatization'' of $CU$? How is (related to) a ``quantization'' of $sl_2$?
\item That funny formula for $R$, wherefore?
\item How do we verify all the Hopf$+R$ properties for $QU$?
\item How do we compute in $QU$?
\item Is this {\bf it}? A poly-time strong tangle invariant, homomorphic for $m$, $\Delta$, $S$?
\end{enumerate}

\end{multicols}

\hrule

\vspace{-3mm}

\begin{multicols}{2}

\begin{theorem} (PBW, Poincar\'e-Birkhoff-Witt) For any Lie algebra $\frakg$ with any choice of ordered basis $\bX=(\bx_1,\bx_2,\ldots)$ with commutative counterpart $X=(x_1,x_2,\ldots)$, the ordering map $\bbO_X\colon\bbQ[X]\to\calU(\frakg)$ defined by $x_1^{n_1}x_2^{n_2}\cdot\ldots \mapsto \bx_1^{n_1}\bx_2^{n_2}\cdot\ldots$ is an isomorphism of vector spaces.
\end{theorem}

\begin{proposition} $c\eta_i = \bbE_{\emptyset\to\{i\}}[0]$, $c\veps^i = \bbE_{\{i\}\to\emptyset}[0]$, $cS_i = \bbE_{\{i\}\to\{i\}}[-\eta_iy_i-\beta_ib_i-\alpha_ia_i-\xi_ix_i]$, $c\Delta^i_{jk} = \bbE_{\{i\}\to\{j,k\}}[\eta_i(y_j+y_k) + \beta_i(b_j+b_k) + \alpha_i(a_j+a_k) + \xi_i(x_j+x_k)]$, and $cm^{ij}_k = \bbE_{\{i\}\to\{j,k\}}[\Lambda]$, where
\begin{multline*} \Lambda =
    \left(\eta _i+\frac{e^{-\alpha _i-\epsilon  \beta _i} \eta _j}{1+\epsilon  \eta _j \xi_i}\right) y_k+\left(\beta _i+\beta_j+\frac{\log \left(1+\epsilon  \eta _j \xi _i\right)}{\epsilon }\right) b_k \\
   + \left(\alpha
_i+\alpha _j+\log \left(1+\epsilon  \eta
   _j \xi _i\right)\right) a_k+\left(\frac{e^{-\alpha _j-\epsilon  \beta _j} \xi _i}{1+\epsilon
\eta _j \xi _i}+\xi _j\right) x_k.
\end{multline*}
\end{proposition}

{\bf $\frac12$Proof.} We show that
\[ \bbe^{\eta_iy_i+\beta_ib_i+\alpha_ia_i+\xi_ix_i +
\eta_jy_j+\beta_jb_j+\alpha_ja_j+\xi_jx_j} \act \bbO_{i,j} \act cm^{ij}_k
= \bbe^\Lambda \act \bbO_k \]
by computing in a faithful 2D representation $\rho$ of $CU$:

\vskip -3.5mm
\def\cellscale{0.645}
\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{../../Talks/UCLA-191101/Snips/FDAExamples/sl2-\the\snip.pdf}
\ifnum \snip<6 \repeat

\end{multicols}

\end{document}


